Simulations of estimating causal effects using undersmoothed HAL
Seraphina Shi
2023-03-17
Simulation #1
Data structure: \(O = (W, A, Z, Y)\)
- U - exogenous variables
- W - baseline covariate that is a measure of body condition
- A - treatment level based on W, continuous between 0 and 5
- Z - intermediate curve based on W and A
- Y - outcome, indicator of an event ?
Underlying data generating process, \(P_{U,X}\)
- Exogenous variables:
- \(U_A \sim Normal(\mu=0, \sigma^2 =
1^2)\)
- \(U_A \sim Normal(\mu=0, \sigma^2 =
2^2)\)
- \(U_Z \sim Uniform(min = 0, max =
1)\)
- \(U_Y \sim Uniform(min = 0, max = 1)\)
- \(U_A \sim Normal(\mu=0, \sigma^2 =
1^2)\)
- Structural equations F and endogenous variables:
- \(W = U_W\)
- \(A = bound(2 - 0.5W + U_A, min=0,
max=5)\)
- \(Z = \mathbf{I}[U_Z < expit(2-W-A)]\)
- \(Y = \mathbf{I}[U_Y < expit(W + 5*A + Z - 0.5 * W * A - 8)]\)
- \(W = U_W\)
## W A Z Y
## Min. :-3.477926 Min. :0.0000 Min. :0.000 Min. :0.0000
## 1st Qu.:-0.677888 1st Qu.:0.6247 1st Qu.:0.000 1st Qu.:0.0000
## Median : 0.000642 Median :1.9837 Median :0.000 Median :1.0000
## Mean : 0.003370 Mean :2.1110 Mean :0.481 Mean :0.5969
## 3rd Qu.: 0.681931 3rd Qu.:3.3781 3rd Qu.:1.000 3rd Qu.:1.0000
## Max. : 3.935166 Max. :5.0000 Max. :1.000 Max. :1.0000## Summary of A given W < -1:## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.000 1.423 2.849 2.753 4.218 5.000## Summary of A given -1 < W <= 0:## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.0000 0.9051 2.2298 2.2719 3.5272 5.0000## Summary of A given 0 < W <= 1:## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.0000 0.4292 1.7748 1.9164 3.1005 5.0000## Summary of A given 1 < W:## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.000 0.000 1.253 1.548 2.649 5.000
##
## Call:
## glm(formula = Y ~ W + A + W * A + Z, family = binomial, data = obs)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -3.4744 -0.0688 0.0008 0.0623 3.7343
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -7.5915 0.2389 -31.775 < 2e-16 ***
## W 0.7056 0.1857 3.800 0.000145 ***
## A 4.7817 0.1352 35.379 < 2e-16 ***
## Z 0.8623 0.1138 7.579 3.48e-14 ***
## W:A -0.3168 0.1210 -2.618 0.008848 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 13485.0 on 9999 degrees of freedom
## Residual deviance: 2719.5 on 9995 degrees of freedom
## AIC: 2729.5
##
## Number of Fisher Scoring iterations: 8## [1] " MSE: 413.5185, AUC: 0.9883"
n = 50
CV HAL
results
## CV selected lambda (from one sample): 0.0195155535952038
1000 repetition
## The average of CV selected lambdas (from 1000 sample): 0.0212969847655493 The average of CV selected lambdas (from 1000 sample): 0.02133982349379 The average of CV selected lambdas (from 1000 sample): 0.0212988488186698 The average of CV selected lambdas (from 1000 sample): 0.0213307345866777 The average of CV selected lambdas (from 1000 sample): 0.0213307345866777## z=1:
## z=0:
Globally Undersmoothed HAL
results
## Undersmoothed lambda: 0.00943173090927282
## which is 1 * lambda_CV
1000 repetition
## [1] " Since the learned undersmoothed lambda is NA, refitting lambdas."
## [1] " Since the learned undersmoothed lambda is NA, refitting lambdas."## The average of unsersmoothed lambda (from 1000 sample): 0.00551793440094463
## which is 1 * the average of 1000 lambda_CV The average of unsersmoothed lambda (from 1000 sample): 0.00549712432627094
## which is 1 * the average of 1000 lambda_CV The average of unsersmoothed lambda (from 1000 sample): 0.0055095193104553
## which is 1 * the average of 1000 lambda_CV The average of unsersmoothed lambda (from 1000 sample): 0.00550188721851507
## which is 1 * the average of 1000 lambda_CV The average of unsersmoothed lambda (from 1000 sample): 0.00550188721851507
## which is 1 * the average of 1000 lambda_CV## z=1:
## z=0:
Oevr a grid of lambda scalers
## TableGrob (7 x 4) "arrange": 7 grobs
## z cells name grob
## 1 1 (2-3,2-3) arrange gtable[layout]
## 2 2 (4-5,1-2) arrange gtable[layout]
## 3 3 (4-5,3-4) arrange gtable[layout]
## 4 4 (6-7,1-2) arrange gtable[layout]
## 5 5 (6-7,3-4) arrange gtable[layout]
## 6 6 (3-3,4-4) arrange gtable[guide-box]
## 7 7 (1-1,1-4) arrange text[GRID.text.2880]
## TableGrob (7 x 4) "arrange": 7 grobs
## z cells name grob
## 1 1 (2-3,2-3) arrange gtable[layout]
## 2 2 (4-5,1-2) arrange gtable[layout]
## 3 3 (4-5,3-4) arrange gtable[layout]
## 4 4 (6-7,1-2) arrange gtable[layout]
## 5 5 (6-7,3-4) arrange gtable[layout]
## 6 6 (3-3,4-4) arrange gtable[guide-box]
## 7 7 (1-1,1-4) arrange text[GRID.text.3156]
## TableGrob (7 x 4) "arrange": 7 grobs
## z cells name grob
## 1 1 (2-3,2-3) arrange gtable[layout]
## 2 2 (4-5,1-2) arrange gtable[layout]
## 3 3 (4-5,3-4) arrange gtable[layout]
## 4 4 (6-7,1-2) arrange gtable[layout]
## 5 5 (6-7,3-4) arrange gtable[layout]
## 6 6 (3-3,4-4) arrange gtable[guide-box]
## 7 7 (1-1,1-4) arrange text[GRID.text.3432]